dimensional analysis of the earthquake-

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Bulletin of EarthquakeEngineeringOfficial Publication of theEuropean Association forEarthquake Engineering ISSN 1570-761XVolume 9Number 2 Bull Earthquake Eng (2011)9:561-579DOI 10.1007/s10518-010-9220-8

Dimensional analysis of the earthquake-induced pounding between inelasticstructures

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Bull Earthquake Eng (2011) 9:561–579DOI 10.1007/s10518-010-9220-8

ORIGINAL RESEARCH PAPER

Dimensional analysis of the earthquake-inducedpounding between inelastic structures

Elias G. Dimitrakopoulos · Nicos Makris ·Andreas J. Kappos

Received: 18 January 2010 / Accepted: 19 October 2010 / Published online: 3 November 2010© Springer Science+Business Media B.V. 2010

Abstract In this paper the seismic response of inelastic structures with unilateral contact isrevisited with dimensional analysis. All physically realizable contact types are captured via anon-smooth complementarity approach. The implementation of formal dimensional analysisleads to a condensed presentation of the response and unveils remarkable order even thoughtwo different types of non-linearity coexist in the response: the boundary non-linearity ofunilateral contact and the inelastic behaviour of the structure itself. It is shown that regardlessthe intensity and frequency content of the excitation, all response spectra become self-similarwhen expressed in the appropriate dimensionless terms. The proposed approach hinges uponthe notion of the energetic length scale of an excitation which measures the persistence ofground shaking to impose deformation demands. Using the concept of persistency which isdefined for excitations with or without distinct pulses, the response is scaled via meaningfulnovel intensity measures: the dimensionless gap and the dimensionless yield displacement.The study confirms that contact may have a different effect on the response displacementsof inelastic structures depending on the spectral region. In adjacent inelastic structures, suchas colliding buildings or interacting bridge segments, contact is likely to alter drastically theexcitation frequencies’ at which the system is most vulnerable. Finally, it is shown that theproposed approach yields maximum response displacements which correlate very well withthe persistency of real earthquakes for a bridge system with considerably complex behaviour.

Keywords Pounding · Unilateral contact · Earthquake engineering ·Non-linear structural dynamics dimensional analysis · Bridges

E. G. Dimitrakopoulos (B)Department of Engineering, University of Cambridge, Cambridge CB2 1 PZ, UKe-mail: [email protected]

N. MakrisDepartment of Civil Engineering, University of Patras, 26500 Patras, Greece

A. J. KapposDepartment of Civil Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

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562 Bull Earthquake Eng (2011) 9:561–579

AbbreviationsDA Dimensional AnalysisLCP Linear Complementarity ProblemMSSS Multi-Span Simply-SupportedPGA Peak Ground AccelerationSDOF Single Degree Of Freedom

1 Introduction

This paper derives from a broader study (Dimitrakopoulos et al. 2009a,b; Dimitrakopouloset al. 2010; Dimitrakopoulos 2010) on the dynamics of pounding between adjacent structuresdue to earthquake shaking. The motivation for the work reported herein, is the need to eluci-date the earthquake response of the pounding bridge segments. This is attempted by bringingforward the fundamental physical similarities that govern the response of such multi-para-metric mechanical systems with the aid of formal dimensional analysis (DA) (Barenblatt1996; Sedov 1992).

In 1995 Malhotra et al. (1995) presented the recorded seismic response of a bridge withjoints, wherein sharp spikes in the acceleration records from the deck were attributed toin-deck pounding. Few years later, Malhotra (1998) made an analytical effort to compre-hend the phenomenon of earthquake-induced pounding in bridges. He concluded that due topounding, column forces as well as, the longitudinal separation at in-deck joints are reduced.His conclusions though, are conflicting those of Jankowski et al. (1998) who studied theimpact of many SDOF oscillators in a row, in an effort to simulate the pounding induced bypropagating seismic waves in multi-span isolated bridges.

Further studies on the impact of bridge segments have been presented by DesRochesand Muthukumar (2002) who examined the impact response of elastic and inelasticoscillators including the event of adjacent structures restrained with cables. That studyconcludes that when the natural frequency and the excitation frequency are separated theone-sided impact is accentuated, whereas impact suppresses the response of oscillators atresonance. At about the same time an analogous study was conducted in Japan by Ruangras-samee and Kawashima (2001) who considered pounding between two linear SDOF oscil-lators and proposed the so-called ‘relative displacement response spectrum with poundingeffect’.

Zanardo et al. (2002) considered the spatial variability of the earthquake excitation whenstudying the response of bridges with pounding. Saadeghvaziri et al. (2000) took into accountthe soil-structure interaction effects on the seismic response of multi-span simply-supportedbridges (MSSS) with pounding phenomena. Dicleli (2008) focused on the ability of elasticgap devices to enhance the performance of pounding seismically isolated bridges under theinfluence of near-fault excitations. Further references on pounding oscillators can be foundin Dimitrakopoulos et al. (2009a,b); Dimitrakopoulos et al. (2010).

The present study builds on the previous work of the authors (Dimitrakopoulos et al.2009a,b; Dimitrakopoulos et al. 2010) which concerned elastic pounding structures andextends the proposed dimensional analysis approach to inelastic structures with inelasticpounding.

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Bull Earthquake Eng (2011) 9:561–579 563

2 Proposed methodology

The methodology presented herein, is rather novel from two different aspects:

(i) In the vast majority of seismic engineering studies impact and contact phenomena aretaken into account via a contact element. Herein, a different approach is adopted thatstems from non-smooth dynamics. All physically feasible unilateral contact configu-rations (impacts, continuous contacts, and detachments) are mathematically treated asinequality problems, namely Linear Complementarity Problems (LCPs), according tomultibody dynamics with unilateral contacts in the form presented by Leine et al. (2003).

(ii) On the other hand, in order to uncover the fundamental physical similarities that describethe pounding behaviour of bridges, formal dimensional analysis (DA) (Barenblatt 1996;Sedov 1992) is implemented. DA is a tool of mathematical physics that shapes thegeneral form of relations that describe physical phenomena. When the behaviour ofpounding structures is described in the dimensionless terms yielded from DA the non-linear, non-smooth response is liberated from the need to refer to a substitute systemand the remarkable property of self-similarity, a special type of symmetry - invariancewith respect to the intensity and the frequency content of the excitation, is revealed. InDimitrakopoulos et al. (2009a,b); Dimitrakopoulos et al. (2010) it was shown that DAoffers a lucid interpretation of the response of elastic pounding oscillators.

2.1 Time and length scales in earthquake records with distinct pulses

The application of the proposed (DA) method hinges upon a distinct time scale (ωg) and alength scale

(ag

)that characterize the ground shaking and are relevant to structural response.

In records with distinct pulses, such time and length scales emerge naturally from the dis-tinguishable pulses which dominate a wide class of strong (usually near-field) earthquakerecords; they are directly related with the rise time and slip velocity of faulting. The minimumnumber of input parameters of such models is two and they have an unambiguous physicalmeaning. Herein the acceleration amplitude αp

(ag = ap

)and duration Tp(ωg = 2π/Tp)of

the pulse are used. Fig. 1 (top) shows the time history from the 1995 Aegion (Greece) earth-quake record, as well as the pulse duration Tp and the pulse acceleration αp .

2.2 Time and length scales in earthquake records without distinct pulses

In many records, including typical Greek records, there are no distinct pulses (Fig. 1 bottom).Nevertheless, it is still feasible to apply the proposed (DA) approach provided the appropri-ate time and length scales are adopted. This is a critical issue of the proposed approach,which is treated in detail in Dimitrakopoulos et al. (2009a,b). In that study such scales,together with the associated selection criteria, are proposed among the available in literaturestrong ground-motion parameters. These are the peak ground acceleration (ag = PG A) aslength scale, and the mean period Tm (Rathje et al. 1998) as time scale (ωg = 2π/Tm),where Tm = ∑

i

(C2

i / fi)/∑

i C2i , Ci are the Fourier amplitudes of the accelerogram and

fi the discrete Fourier transform frequencies between 0.25 and 20 Hz. It is recalled here,that when the focus is on a peak response parameter, it is the frequency content and not theduration of an excitation that is of interest (see Dimitrakopoulos et al. (2009a,b) and ref-erences therein). When the response is described in dimensionless terms that hinge uponthose parameters (PG A, Tm), the remarkable order of self-similarity emerges. In addi-tion, this approach reduces drastically the scatter in the response of some fundamentalmechanical configurations of earthquake engineering (Dimitrakopoulos et al. 2009a,b): the

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Fig. 1 Time and Length Scales in earthquake records with (top) and without (bottom) distinct pulses

elastic-plastic system and the pounding oscillator. Herein the same approach is extended toinelastic structures with inelastic pounding.

3 Dimensional analysis of an elasto-plastic pounding system

In general, the equation of motion of an inelastic SDOF oscillator, subjected to base excitationand taking into account contact phenomena can be written as:

mu(t) + Fint(t) − W · λ(t) = −mug(t) (1)

where u is the relative, to the ground, response displacement, ug is the ground displacement,m the mass, and Fint , the internal inelastic force of the oscillator. W is the direction vectorof the constraint (contact) force λ which herein is considered as a Lagrange multiplier, seealso Dimitrakopoulos et al. (2009a,b).

In case of elasto-plastic behaviour the internal inelastic force Fint is given by:

Fint(t)

m= 2ξωs · u(t) + Q

m· z(t) (2)

where ξ is the damping ratio, ωs the angular frequency of the structure after yielding (post-yielding), Q the specific strength of the structure and z is a dimensionless hysteretic parameter,with |z| ≤ 1, that is given by:

z(t) = 1

uy

[u(t) − γ |u(t)| z · |z|n−1 − βu(t) |z|n]

(3)

Equation (3) is a special case of the Bouc-Wen model (Wen 1975, 1976) that has been usedextensively for simulating seismic isolation systems (Makris and Chang 2000). Parametersβ, γ , and n are dimensionless quantities that shape the hysteretic loop and herein are takenas: β = γ = 0.5 and n = 20 (Makris and Chang 2000).

Substituting Eq. (2) into the equation of motion (1) yields:

u(t) + 2ξωs · u(t) + (Q/m) · z(t) − (1/m) · W · λ(t) = −ug(t) (4)

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Conclusively, the parameters necessary to determine a response quantity of an elasto-plasticpounding oscillator subjected to ground excitation, e.g. the maximum response displacementumax are: uy the yield displacement and Q/m the characteristic strength, the accelerationamplitude αg and the angular frequency of the excitation ωg (Fig. 1), the initial distancebetween the oscillator and the rigid barrier “gap” δ, and the coefficient of restitution εN.Assuming a pure hysteretic behaviour (zero viscous damping ξ = 0) the response functioncan be written as:

umax = f (uy, Q/m, δ, εN , ag, ωg) (5)

This results in a group of 7 variables which involve only 2 reference dimensions, those oflength [L] and time [T]. According to Buckingham’s “�” theorem the number of independentdimensionless �-products is now: (7 variables) − (2 reference dimensions) = 5 �-terms.

Herein, the characteristics of the excitation, αg and ωg are selected in order to normalisethe non-linear response (including both yielding and pounding) to the energetic length scale ofthe excitation, Le = αg/ω

2g The product αg/ω

2g is a characteristic length scale of the (ground)

excitation’s persistence to impose deformation demands, see for instance Dimitrakopouloset al. (2009a,b) and references therein. Accordingly, Eq. (5) reduces to:

umaxω2g

ag= φ

(δω2

g

ag, εN ,

uyω2g

ag,

Q

mag

)

(6)

or

�um = φ(�δ,�ε,�uy,�Q) (7)

with:

�um = umaxω2g

ag,�δ = δω2

g

ag,�ε = εN ,�uy = uyω

2g

ag,�Q = Q

mag(8)

The dependent variable, dimensionless product �um, is the maximum response displacementnormalized to Le. The dimensionless gap �δ and the coefficient of restitution εN are thepounding parameters, whereas the dimensionless yield displacement �uy and the character-istic strength �Q are associated with the elasto-plastic behaviour of the structure. The dimen-sionless gap �δ = δω2

g/αg and the dimensionless yield displacement �uy = uyω2g/αg are

novel intensity measures which suggest that the size of the gap “δ” and the yield displacementuy accordingly, can be scaled to the energetic length scale Le=αg/ω

2g [m] (Dimitrakopoulos

et al. 2009a,b). In contrast to other intensity measures proposed recently in the literature(Vega et al. 2009), �δ and �uy are rationally produced and dimensionally consistent inten-sity measures which as illustrated later on, correlate well with the response of such mechanicalconfigurations.

Figure 2 presents the response spectra of an elasto-plastic pounding system to a cosinepulse, for different excitation intensities. The most decisive feature of the proposed DAapproach (dimensionless terms �δ and �uy) is that it brings forward the property of self-similarity even when the two different types of non-linearity coexist, i.e. the unilateral contactphenomena and the inelastic behaviour of the structure itself. The response curves expressedin the �-terms for different intensity and frequency of the excitation follow a master-curve(self-similarity—Fig. 2 right)

Figure 3 presents simultaneously the two key parameters of the proposed approach: thedimensionless gap �δ = δω2

g/αg and the dimensionless yield displacement �uy = uyω2g/αg .

Figure 3 left plots contours of �δ on the plane: (δ)− (1/Le) for several types of bridges and

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Fig. 2 The response spectra for different excitation intensities and a given dimensionless gap �δ , and yielddisplacement �uy (left) collapse to a single curve (self-similarity) when expressed in the proposed dimen-sionless �-terms (right)

Fig. 3 Left: Contours of dimensionless gap �δ = δω2g/αg values, for several bridge types and earthquakes.

Right: Contours of dimensionless yield displacement �uy = uyω2g/αg values, for several seismic isolation

types and earthquakes. The vertical axis in both diagrams scales the inverse of the earthquake’s persistency(1/Le = ω2

g/αg)

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earthquakes. Typical values of in-deck expansion joints range from 0.6 to 1.3 cm (DesRochesand Muthukumar 2002), in MSSS bridges (Saadeghvaziri and Yazdanimotlagh 2008) from2.5 to 7.5 cm while larger values are common in deck-abutment expansion joints. Figure 3(right) plots contours of �uy on the plane: (uy) − (1/Le) for several seismic isolation types.Representative values of uy for each seismic isolation type are drawn from Makris and Black(2004a). The vertical axis is common for both diagrams (left and right) of Fig. 3 and scalesthe inverse of the excitation’s persistency (1/Le = ω2

g/αg). Greek earthquakes records aresystematically less persistent (Fig. 3) than popular earthquakes records as for instance theones considered in Makris and Psychogios (2006). As a consequence, the same gap size δ

corresponds to a smaller dimensionless gap �δ when the bridge is subjected to an earthquakelike San Fernando (1971) (Makris and Psychogios 2006) than to an earthquake like Lefkada2003 (Fig. 1 bottom).

The shaded areas, in both diagrams under the contour �δ=0.1(and �uy = 0.1 accord-ingly) correspond to the area where complete similarity prevails. The property of completesimilarity reveals that the response of a pounding oscillator is indifferent to small values ofthe dimensionless gap (�δ < 0.1) (Dimitrakopoulos et al. 2010) and that the response of anelasto-plastic system is indifferent to small values of the dimensionless yield displacement(�uy < 0.1) (Makris and Black 2004b) accordingly. For instance, if the dimensionless yielddisplacement �uy turns out small enough (�uy < 0.1) the engineer can expect that the initialpre-yielding stiffness may be immaterial to the actual response of the structure (Makris andBlack 2004b). What is also evident from Fig. 3 is the similar way that dimensionless terms �δ

and �uy depend on the earthquake’s persistency. Thus, the estimation of the dimensionlessterms describing the response (e.g. �δ and �uy after evaluating Le) may provide, throughdiagrams like Fig. 3, information on the sensitivity of the structure to phenomena like yieldingor pounding before an analysis is performed.

Figure 4 illustrates the response spectra for a given dimensionless yield displacement(�uy = 0.2) and several dimensionless gaps �δ , of an elasto-plastic SDOF oscillator sub-jected to pulse-type excitations. The response spectra of Fig. 4 are self-similar; they areindifferent to the excitation’s intensity and period. Similar results are obtained for severaldimensionless yield displacements e.g.�uy = 0.01, even though they are not presented hereinfor economy of space. Self-similarity is a well-known property of linear elastic response,hence response curves for the no-contact case (grey lines in Fig. 4) illustrate the same feature.

The accentuation of the response, due to contact, is intense in the area of weak sys-tems/short pulses (small �Q values). The opposite holds true for intermediate �Q values,the response diminishes due to contact. In addition, in the area of strong systems/ longpulses (high �Q values) the maximum response displacement is indifferent to contact. Con-sequently, three distinct spectral areas are recognized in the response, which appear system-atically for inelastic as well as linear elastic pounding oscillators (Dimitrakopoulos et al.2009a,b). Compared with the bilinear behaviour (Fig. 5), the accentuation of the responsedue to contact is more pronounced in the case of elasto-plastic behaviour, which is attributedto the (comparatively) lower restoring force.

The examined mechanical configuration lacks symmetry with respect to the direction(normal/reverse) of motion. Left and right columns of Fig. 4 differ only in the directivity ofthe pulse and reflect this peculiarity. Differences in the response are more pronounced forexcitations with a strong “preference” in a specific direction like sine pulses, which repre-sent pure forward motions. The directivity effects wear off for excitations with more loadingcycles and/or no preference to a specific direction. In modern literature lack of symmetryin configurations of pounding structures is often quite disregarded with the risk of loosingvaluable information (Dimitrakopoulos et al. 2010).

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Fig. 4 Self-similar response spectra of the elasto-plastic pounding system, subjected to pulse-type excitationsof normal (left) and reverse (right) directivity

Depending on the excitation’s shape and directivity, response may be very sensitive tomoderate and high dimensionless gap values (Fig. 4). However, for small enough dimension-less gaps (�δ < 0.5) the response displacement is indifferent to �δ . This invariance of theresponse for small �δ values confirms the existence of the complete similarity region (greyarea) in Fig. 3 for inelastic structures, originally observed for the elastic pounding oscillator(Dimitrakopoulos et al. 2010).

4 Dimensional analysis of a bilinear sdof pounding oscillator

Before moving on to bilinear pounding structures, it is useful to revisit first inelastic struc-tures without pounding. The dynamic equilibrium of a bilinear SDOF oscillator with massm that is subjected to a ground excitation gives:

u(t) + Fint(t)

m= −ug(t) (9)

with the same notations as in Eq. (1).

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Fig. 5 Self-similar response spectra of a bilinear pounding structure, subjected to pulse-type excitations ofnormal (left) and reverse (right) directivity

A bilinear hysteretic loop (Fig. 5 top) can de defined with an appropriate combinationof three of the following parameters: characteristic strength Q, yield force Fy , pre-yield-ing stiffness K0, post-yielding stiffness (second slope of the bilinear loop) Ks , hardeningα = Ks/K0, and yield displacement uy . Consequently, there is no unique way, but depend-ing on the adopted parameters, several alternative ways of describing a structure with bilinearbehaviour.

Seismically isolated structures are designed intentionally with small yield displacementsand/or strength in order to perform inelastically when subjected to the design earthquake.Makris and Black (2004a), expressed the response of isolated structures with the triplet:Q(or Q/m), uy and ωs(or Ts), and revealed the remarkable property of complete similaritywith respect to small dimensionless yield displacements �uy. This property unveils that theelastic characteristics of structures designed with small yield displacements are indifferent tothe (essentially inelastic) response. The response function of a bilinear SDOF pounding oscil-lator (Fig. 5 bottom), when the inelastic behaviour is described with the triplet (Q or Q/m,

uy and ωs or Ts), can be written as:

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umax = f

(ωs,

Q

m, uy, δ, εN , ag, ωg

)(10)

and by following an analogous analysis, the relation of the dimensionless �-terms is:

�um = φ(�ωs,�δ,�ε,�uy,�Q) (11)

with:

�ωs = ωs

ωg,�Q = Q

mag(12)

and the remaining �-terms are as in Eq. (8).Figure 5 presents the same results as Fig. 4, but for a bilinear instead of elasto-plastic,

pounding SDOF system. The bilinear behaviour is described in terms of the dimensionless�ωs = ωs/ωg while the elasto-plastic behaviour is expressed in terms of �Q = Q/mαg andappears as the limit case of the bilinear one for �ωs = 0 The self-similar response spectraof Fig. 5 illustrate the same trends (as Fig. 4) regarding the influence of contact.

Figure 6 plots the self-similar response spectra of an SDOF pounding oscillator withelastic (top), elasto-plastic (middle) and bilinear (bottom) behaviour. In all three cases thedimensionless gap �δ , the coefficient of restitution εN and the excitation are the same. Thedimensionless approach proposed herein unveils the property of self-similarity in all casesconsidered, even though the response is strongly non-linear (non-smooth) due to contactas well as inelastic (elasto-plastic or bilinear). What is also noteworthy about Fig. 6 is thequalitatively similar way in which contact influences the structural response. Whether thestructural response is elastic, elasto-plastic or bilinear, the spectrum shape resembles anasymptotic (exponential) form and the aforementioned three spectral areas appear. In otherwords, when the dimensionless gap is small enough, contact alters the response up to thepoint that it overshadows its original characteristics.

5 Multiple inelastic pounding oscillators

In the general case, a mechanical configuration of n bilinear SDOF oscillators in a row isgoverned by 4n + 4 variables. These are:

• ωsi , uyi , Qi , and mi , and mi : the post-yielding angular frequency, the yield displace-ment, the characteristic strength and the mass of the oscillator with index i , where i = 0÷ n − 1. In the following, the oscillator under examination is denoted with sub-index 0and the equivalent viscous damping ratio of the oscillators is ignored (ξ = 0).

• εN and δ: the coefficient of restitution and the gap.• ag and ωg: a characteristic length and time scale of the excitation (Fig. 1)

Hence a response quantity of interest, e.g. the maximum response displacement umax , of anoscillator, can be written as a function of the general form:

umax = f(ωsi , Qi , mi , uyi , δ, εN , ag, ωg

), i = 0 ÷ n − 1 (13)

This results in a set of 4n + 5 variables which involve 3 reference dimensions, those oflength [L], time [T] and mass [M]. According to Buckingham’s “�” theorem the numberof independent dimensionless �-products is now: (4n + 5 variables) – (3 reference dimen-sions) = 4n + 2�-terms.

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Fig. 6 Response spectra of a SDOF pounding oscillator. elastic (top), elasto-plastic (middle) and bilinear(bottom)

Extending the same approach, the characteristics of the excitation, αg and ωg , and theproperties of a reference-oscillator (ω0 and m0) are selected as repeating variables.Accord-ingly, Eq. (5) reduces to:

umaxω2g

ag= φ

(δω2

g

ag, εN ,

ωs0

ωg,

uy0ω2g

ag,

Q0

m0ap,

m0

mi,ωs0

ωsi,

uyiω2g

ag,

Qi

mi ag

)

, i = 1 ÷ n − 1

(14)

or

�um = φ(�δ,�ε,�ωs0,�uy0,�Q0,�mi,�si,�uyi,�Qi), i = 1 ÷ n − 1 (15)

with:

�ωs0 = ωs0

ωg,�si = ωs0

ωsi,�mi = m0

mi(16)

and the remaining �-terms as in Eq. (8).

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Fig. 7 Self-similar response spectra of a pair of bilinear pounding structures, subjected to pulse-type exci-tations. Left column corresponds to most flexible and right column to most stiff among a pair of structures

Eq. (14) shows that 10 �-terms govern the problem of a pair (n = 2) of bilinear SDOFstructures (Fig. 7) and 14�-terms the response of three (n = 3) bilinear structures in a row(Figs. 8 and 9). Thus, Eq. (14) clearly illustrates the multiparametric nature of the dynamicsof pounding-structures.

For convenience, simplifying assumptions are made in the following parametric study.It is assumed that the mass, strength and yield displacement of the structures examined arethe same: m0 = mi , Q0 = Qi , uy0 = uyi . Under these rather restrictive assumptions Fig. 7plots the response spectra of a structure (with characteristics: ω0, m0, uy0 and Q0) involvedin a pair of bilinear structures. Figures 8 and 9 present the same results as Fig. 7 but for thecentric structure, from a symmetric triplet of bilinear pounding structures. Left columns (ofFigs. 7, 8, 9) present the response of the more flexible structure of the system considered(�s1 < 1), while right columns present the response of the stiffer structure (�s1 > 1).

As the (post-yielding) angular frequencies of the structures separate, the influence of con-tact becomes more pronounced (compare Fig. 8 with 9), in agreement with the pertinentconclusions of DesRoches and Muthukumar (2002). It is reminded though, that this is trueprovided the masses of the colliding structures are the same. A more detailed analysis ofthe role of the mass ratio shows that when masses differ substantially contact may alter

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Fig. 8 Self-similar response spectra of a symmetric triplet of bilinear pounding structures, subjected to pulse-type excitations. Left column corresponds to most flexible and right column to most stiff structure

drastically the response of linear pounding oscillators even if structural frequencies are sim-ilar (Dimitrakopoulos et al. 2009a,b).

As a result of contact, the shape of the spectrum of a flexible structure sharpens, while thatof a stiff structure flattens. This characteristic is associated with the three spectral regions,previously noted for single pounding structures, which arise also for the considered systemsof multiple pounding structures. The difference is that the response of flexible structures isaccentuated in the area of short pulses (small �ωs), while the response of stiff structuresis accentuated in the area of long pulses. For �ωs near unity the maximum response dis-placement usually diminishes for all structures. A collateral consequence is that in adjacentinelastic structures, such as individual deck segments or neighbouring buildings, contactalters the frequencies which stimulate the most the individual parts of the system comparedwith the no-contact dynamic behaviour. For instance when a structure is excited near its effec-tive (post-yielding) frequency its response may be hindered due to contact, but at the sametime the response of their neighbouring structure is often accentuated. This is a critical trendfrom a practical point of view, observed also in linear pounding oscillators (Dimitrakopouloset al. 2009a,b), since contact alters the excitation frequencies’ at which such mechanicalconfigurations are most vulnerable.

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Fig. 9 Self-similar response spectra of a symmetric triplet of bilinear pounding structures, subjected to pulse-type excitations. Left column corresponds to most flexible and right column to most stiff structure

The proposed dimensionless spectra (Figs. 7, 8, 9) are self-similar, i.e. (for given valuesof the associated �-terms) are indifferent to the intensity of the excitation. Self-similarityis of unique importance since it reveals invariance with respect to scale transformations(intensity of the excitation). At the same time, traditional response spectra, useful for spec-trum-based design, can be calculated from the proposed spectra through a simple similaritytransformation.

6 Illustrative example

The proposed methodology is implemented on a two-segment frame bridge (Fig. 10) withthe characteristics presented in Table 1, similar with those in DesRoches and Muthukumar(2002).

The equation of motion for the bridge system examined, taking into account unilateralcontact can be written as:

Mu + FF (u) − FB(u j − ui ) − Wλ = −Mδug (17)

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Fig. 10 Top: Layout of the bridge examined. Middle: Mechanical model of the bridge system and parametersinvolved. Bottom: non-linear elements of the model

Where the mass matrix is: M = diag {m1 m2} , λ is the contact force considered as a Lagrangemultiplier (see also Dimitrakopoulos et al. (2009a,b), δ is the unit vector and W the directionvector. The restoring characteristics (FF ) of the two segments of the bridge are modelledwith pertinent bilinear hysteretic Bouc-Wen springs (Fig. 10—bottom left):

FF (u) =[

FF1 (u1)

FF2 (u2)

]=

[ks1u1 (t) + Qs1zs1 (t)ks2u2 (t) + Qs2zs2 (t)

](18)

where ks denotes the post-yielding stiffness, Qs the specific strength and z the dimensionlesshysteretic parameter, governed by Eq. (3). A ξ = 5% viscous damping ratio and a coefficientof restitution εN = 0.7 is assumed.

Table 1 Characteristics of the bridge examined

Element Initial stif. (kN/m) Yield str (kN) Period (s) Strain hardening Weight (t)

Frame − F1 2,33,443 3,621 0.47 5% 1,301

Frame − F2 1,01,048 3,794 1.12 5% 3,210

Element Effective stif. (kN/m) Element Active stif. (kN/m) Passive stif. (kN/m)

Bearing − B1 1,051 Abutment A1, A2 1,751 4,55,328

Bearing B0, B2 1,50,000

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0 5 10 15 20 25 30 35 40 45-10

0

10

0 5 10 15 20 25 30 35 40 45

0

5

10

-0.04 -0.02 0 0.02-200

-100

0

100

200

-0.05 0 0.05-200

-100

0

100

200

uωg2 /α

g

tωg /(2π)

FF1

u1

FF2

u2

g Nω

g2 /αg

Fig. 11 Earthquake response analysis for the Lefkada 2003 record (shown in Fig. 1 bottom). Time historiesin dimensionless terms: of the displacements of the two segments illustrating their relative distance gN (top)and their relative distance gN (middle). Bottom: the force—displacement loops of the two segments

The bearings are modelled using linear springs approximating their effective stiffness(FB):

FB(u) =[

FB1 (u2 − u1) − FB0 (u1 − u0)

FB2 (u3 − u2) − FB1 (u2 − u1)

]

=[

kb1 (u2 − u1) − kb0 (u1 − u0)

kb2 (u3 − u2) − kb1 (u2 − u1)

]

(19)

The behaviour of the abutments is assumed to follow the elastic bilinear spring (FA) shownin Fig. 10 (bottom right).

The bridge of Fig. 10 is analysed for the 62 Greeck records considered in Dimitrakopouloset al. (2009a,b), comprising practically most of the available historic Greek records, with-out reference to any substitute pulses. Instead, all records are characterised directly usingαg = PG A and ωg = 2π/Tm , where Tm is the mean period (Rathje et al. 1998). With theaforementioned scales the notion of persistency or energetic length scale Le = αg/ω

2g [m] is

extended to excitations with or without distinct pulses.Sample results of the earthquake response analyses are offered in Fig. 11 for the Lefkada

2003 record of Fig. 1-bottom. In Fig. 11 open circles denote impacts, while filled circlesdenote the end of a continuous contact (detachment). The at-rest size of the joint between thetwo segments is: δ = 1.27 cm, hence the dimensionless gap term for (typical) Greek records(Dimitrakopoulos et al. 2009a,b) is estimated as �δ = δω2

g/αg > 0.5. Similarly, the yielddisplacement of the two segments is accordingly uy1 = 0.016 and uy2 =0.038 m, so thedimensionless yield displacements are respectively greater than �uy1 = uy1ω

2g/αg > 0.5

and �uy2 > 1.2.

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0 0.005 0.01 0.015 0.02 0.025 0.030

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

yielding

0 0.005 0.01 0.015 0.02 0.025 0.030

0.01

0.02

0.03

0.04

0.05

0.06

Tg<0.2sec

0.2<Tg<0.40.4<Tg<0.6

Tg>0.6sec

Stiff Frame

u max

1 [m

]

yielding

Flexible Frame

u max

2 [m

]

Le = ag/ωg2 [m]

Tm ≤ 0.2 sec 0.2 < Tm ≤ 0.4 0.4 < Tm ≤ 0.6 Tm > 0.6 sec

Tm ≤ 0.2 sec 0.2 < Tm ≤ 0.4 0.4 < Tm ≤ 0.6 Tm > 0.6 sec

62 Greek records

62 Greek records

Fig. 12 Maximum response displacement of the stiff (top) and the flexible frame (bottom) vs. the energeticlength scale of 62 Greek records (Dimitrakopoulos et al. 2009a,b)

The results from the 62 records (Fig. 12) reveal a strong correlation between the energeticlength scale Le (persistency) of (real) excitations with the maximum response displacementumax . This is true, despite the complexity of the system’s behaviour and the fact that two exci-tations with the same αg and ωg may differ substantially in (the accelerogram) shape whichinevitably introduces a scatter in the response. For comparison, all excitations are groupedinto 4 sets depending on their mean period Tm (Dimitrakopoulos et al. 2009a,b). Even though(mean) periods as well as the peak ground acceleration of each record may differ substan-tially, the response scales well with persistency Le. Thus, the proposed DA approach verifiesthat what matters in the response of even more complex mechanical configurations are thedimensionless intensity measures, dimensionless gap �δ and yield displacement �uy values,and consequently the persistency of the excitation Le. At the same time, plots like Fig. 12 canbe useful, from a practical standpoint, in the same sense that traditional incremental dynamicanalysis curves are.

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7 Conclusions

The present paper proposes a novel, alternative way to describe the behaviour of pound-ing inelastic structures by implementing formal dimensional analysis in an effort to identifydistinct physical similarities. Pounding structures comprise multi-parametric mechanical sys-tems with quite complex earthquake response for which conflicting conclusions have beenpublished in literature.

The application of the proposed method hinges upon the notion of persistency, or energeticlength scale, of an excitation that assesses the tendency of ground shaking to impose defor-mation demands. The concept of persistency which is defined for excitations with or withoutdistinct pulses is exploited herein to scale the response via meaningful new dimensionlessintensity measures. The proposed approach leads to a condensed presentation of the responseand most importantly unveils the remarkable property of self-similarity even though two dif-ferent types of non-linearity coexist in the response: the boundary non-linearity of unilateralcontact and the inelastic behaviour of the structure itself. It is shown that when the responseis expressed in the appropriate dimensionless �-terms, response spectra for any intensityand frequency content of the excitation collapse to a single master curve (self-similarity).

All physically realizable contact types are captured via a non-smooth complementarityapproach. The present analysis also concludes that contact may have a different effect on theresponse displacements of inelastic structures depending on the structural and excitation’sfrequency ratio (spectral region). As for linear pounding oscillators, three distinct spectralregions are identified herein for inelastic pounding structures. The study also confirms thatin adjacent inelastic structures, such as colliding buildings or interacting bridge segments,contact is likely to alter drastically the excitation frequencies at which the system is mostvulnerable. Most often, contact results in amplifying the response of the most flexible, amongpounding structures, in the low range of the frequency spectrum and the response of the moststiff structure in the upper range of the frequency spectrum.

Finally, the study concludes that the proposed approach yields maximum response dis-placements which correlate very well with the persistency of real earthquakes for a bridgesystem with considerably complex (non-linear and non-smooth) behaviour.

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dimensional analysis of the earthquake-

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